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Unfolding Methods
Alberto RotondiUniversità di Pavia
Folding is a common process in physics
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signal Apparatusresponse
Observed signal
xyxxfygyxxf d),()()(),()(
x
Convolution is a linear folding
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signal Apparatusresponse
Observed signal
xxyxfygxyxf
xzyxyz
d)()()()()(
x
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auxiliary variable
The jacobian is
From the general theorem one obtains
by integrating on the auxiliary variable
hence
which is the probability density
Foldingtheorem
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Convolutiontheorem
For independent variables:
when Z is given by the sum
we have
and we obtain
When X1 and X2 are independent, we obtain the convolution integral
In physics
( instrument function, f signal)
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Uniform*GaussianWhen where
one has immediately
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values
= R* +
1D Unfolding
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2D Unfolding
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Fourier Techniques
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Image Deconvolution
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original
Poissonstatistics
Gaussiansmearing
Fourier(un)restored
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original Poissonstatistics
Gaussiansmearing
Fourierrestored
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original
Poissonstatistics
Gaussiansmearing
Fourier(un)restored
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original Poissonstatistics
Gaussiansmearing
Fourierrestored
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original
Poissonstatistics
Gaussiansmearing
Fourier(un)restored
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La PET: tomografia a positroniPET: positron emission thomography
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Imagerestoration
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Explanation:
The smeared distributions of two input distributions cannot be distinguished if they agree on a large scale of x but differ byoscillations on a “microscopic” scale much smaller than the experimental resolution
or
to increase the DoF by using
a parametric model
)'|()()|( PPP
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1)(P
The frequentistassumes
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Regularizationterms
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What isMaxEnt ???
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The iterative principle
)
(26)
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The iterative Principlewithoutbest fit
Good!
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The iterative Principlewithoutbest fit +smoothing
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The iterative Principlewithoutbest fit
Bad!
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The iterative algorithm +best fit
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The iterative algorithm +Best fit
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The iterative algorithm +best fit +regularization
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The iterative algorithm +best fit +MaxEntregularization
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The iterative algorithm +best fit +Tichonovregularization
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The iterative algorithm +best fit +Tichonovregularization
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The iterative algorithm +best fit +Tichonovregularization
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ATHENA experimental set-upATHENA apparatus
Silicon micro
strips
CsI
crystals
511 keV
511 keV
Charged tracks to reconstruct antiproton
annihilation vertex.
Identify 511 keV photons from e+-e-
annihilations.
Identify space and time coincidence of the
two with ± 5 mm and 5 s resolution
(Probability of a random coincidence:
0.6% per pbar annihilation without
considering detection efficiency)
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From the ATHENA detector
Pbar-only (with electrons)
x
y
cm
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antihydrogen !!!!!!!!!! FIRST COLD ANTIHYDROGEN PRODUCTION & DETECTION (2002)M. Amoretti et al., Nature 419 (2002) 456M. Amoretti et al., Phys. Lett. B 578 (2004) 23
SIGNAL ANALYSIS:
opening anglexy vertex distribution
radial vertex distribution
65 % +/- 10% of annihilations
are due to antihydrogen
between 2002 & 2004 more than 2 millionsantihydrogen atomshave been produced
that’s about 2 x 10-15 mg.. or .. 1000 Giga years for a gram
8110
80;5.6
190
80;7.4
110190
80
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Cold Mix dataHbar (MC) BCKG
(HotMixData)
Pbar vertex XY projection (cm)
x = 0.65 ± 0.05
x Hbar + (1-x) BCKG = Cold Mix
Hbar percentage
Annihilation vertex in the trap x-y plane
ML Fit Result
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The vertex algorithm resolution function is gaussian with
mm3
The 2D deconvolution revealstwo different annihilation modes
Cold Mix
exp background
Cold Mix dataIteratve best fit method
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The iterative algorithms +best fit + regularization
• iterative algorithms are used in unfolding (ill posed) problems
• they need a Bayesian regularization term
• when there are degrees of freedom,onecan use a best fit of a signal+background function to the data
• in this case there are no Bayesian terms(pure frequentist approach)
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end
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Regularizationparameter
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Conclusions• best fit minimization methods are crucial in physics. They are mainly frequentist
• they are based on the ML and LS algorithms (they are implemented in the ROOT-MINUIT framework)
• to judge the quality of the result,frequentists use the testbayesians use the hypothesis probability
• Bayesian a priori hypotheses should be used withinformative priors!!!
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Imagerestoration
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2D Unfolding
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The iterative principle
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Unfolding techniques
Statistica III
